Question

The histogram shows the neck sizes (in inches) of 250 men recruited for a health study in Utah. Which summary statistics would you choose to summarize the center and spread in these data? Why?

Short answer

Use mean and standard deviation for symmetric data; use median and IQR for skewed data.

Step-by-step solution

Identify the Center of the Data

To summarize the center of the data, we can use either the mean or the median. The mean is the average value, calculated by adding all the values and then dividing by the number of values. The median is the middle value when all values are ordered from smallest to largest. If the data is symmetric, the mean is appropriate; if the data is skewed, the median is better.

Determine the Spread of the Data

The spread of the data can be summarized using measures like standard deviation or interquartile range (IQR). Standard deviation is most useful for symmetric data as it shows how much the data varies from the mean. IQR, the range between the first and third quartiles, is better for skewed data as it is not affected by outliers.

Analyze the Shape of the Histogram

Examine the histogram to determine the distribution's shape. If the histogram is symmetric and has a normal distribution shape, then the mean and standard deviation would be appropriate measures. If the histogram is skewed, then the median and IQR should be used.

Choose Summary Statistics Based on Shape

If the histogram shows a normal distribution, choose the mean and standard deviation to summarize the center and spread. If the histogram is skewed, choose the median and IQR as they provide more accurate descriptions of the central tendency and variability in the presence of skewness.

Key concepts

Mean vs. Median

When we talk about the center of a dataset, we are often referring to its central tendency. The mean and median are two popular measures to understand this.

• **Mean**: This is simply the arithmetic average. You add up all the values and then divide by the total number of values. It's a great measure if your data is symmetrically distributed and doesn't have outliers.
• **Median**: This represents the middle point. When you sort all the data points, the median is the one right in the center. It's better than mean for skewed data because it isn't influenced by extremely high or low values.

Using the mean for symmetrical data makes sense because every single point has nearly equal influence. However, if your data has a few unusual points, the median will give a more realistic picture of the center.

Standard Deviation

Standard deviation is a measure that tells us how spread out the numbers in a dataset are around the mean. It's a crucial concept in statistics that helps in understanding variability.

• **Calculation**: To compute it, you first find the difference between each data point and the mean, square these differences, obtain the average of these squares, and finally take the square root of this average.
• **Interpretation**: A smaller standard deviation means data points are close to the mean while a larger one indicates more spread out data.

Standard deviation is ideal when working with symmetric distributions and when comparing variability while assuming data is normally distributed. This makes it a go-to when describing datasets that fit these criteria.

Interquartile Range (IQR)

The interquartile range (IQR) is a measure of statistical dispersion. It gives an idea about how data is spread in the middle half of the dataset and is especially useful in datasets with outliers or skewness.

• **Computation**: To find the IQR, you subtract the first quartile (25th percentile) from the third quartile (75th percentile). This range informs you about the middle 50% of your data.
• **Robustness**: Unlike standard deviation, the IQR isn't affected by outliers. This makes it a robust measure of spread particularly useful for skewed distributions.

In datasets where the presence of outliers or skewedness might distort the understanding of variability, the IQR is preferred to maintain the integrity of the analysis.